Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum

TL;DR

This paper explores lifting the $rak{sl}_2$ action on annular Khovanov homology to a stable homotopy level, revealing new interactions with Steenrod algebra and developing advanced techniques for moduli space analysis.

Contribution

It introduces a method to lift the $rak{sl}_2$ action to spectra and develops new technical tools for analyzing moduli spaces in the framed flow category.

Findings

$rak{sl}_2$ action on homology commutes with Steenrod algebra

Lifting of $rak{sl}_2$ action to spectra achieved

New techniques for cancellations in cube of resolutions

Abstract

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.